THE u-v METHOD 



137 



Additional lobe points are located by choosing 

 some other value of sin- (0/2), say sin^ (S2/2) = 0.7. 

 Each value of sin- (12/2) now gives two points on each 

 lobe, one on the upper branch and the other on the 

 lower. Again choose values oi K = D, obtain the 

 corresponding values of the radical 



and from equation (19) 



V 



(1 - K)- + 4A' sin= 







from Figure 12 in Chapter 5, and calculate new' 

 values for v. The D-v values are plotted as crosses 

 on Figure 7 and the line through them is the locus of 

 points for which sin^ (t2/2) = 0.7 (see Table 3). 



Table 3. Values of v and R for sin^ (.2/2) = 0.7, (b = 2. 



For sui2 (0/2) = 0.7, sm (0/2) = dbO.836, 0/2 

 = 0.315x or 0.6857r. Then = »i7r = 0.637r + 2fc7r 

 and 1.37Tr + 2kir, in which k is an integer. Then 

 n = 0.63 + 2fc and 1.37 + 2k, and R = nr. Values 

 of n and R are listed in Table 3. The intersections 

 of the R values and the locus for sin- ( 0/2) = 0.7 are 

 plotted as circles on Figure 7. 



The entire lobe structure for one contour may be 

 drawn by choosing additional values of sin- (0/2). 

 A large number of contours have been calculated 

 by the Radiation Laboratory and are plotted in 

 Figures 16 to 39. 



In order to construct one contour of a coverage 

 diagram, it remains to find the intersection between 

 the curves giving values of u for constant sin- (0/2) 

 and the corresponding path-difference contours. 

 The equations relating R to 0/2 are given below. 

 From equation (18) 



:© 



From equation (83) in Chapter 5 



fcaA 

 K = . 



hidT 



An assigned value of fixes two values of 5 for each 

 lobe, as explained in the previous paragraph. All 

 values of sin- (0/2) other than 1 or determine two 

 intersections with the lobe. When sin- (0/2) = 1, 

 the envelope of maxima is obtained, while sin- (0/2) 

 = corresponds to the envelope of minima. 



By selecting several values of sin- (0/2) in Figure 

 12, Chapter 5, and following the method outlined 

 above, a coverage diagram may be constructed in 

 generalized {u,v) coordinates. The actual values of 

 hi and d are 



h = hiu, (24) 



d = drv. (25) 



6.5.3 



Construction of Lobes 

 (Vertical Polarization) 



Problems involving vertical polarization or cases 

 where the ratio of the antenna-pattern factors 

 Fi/Fi cannot be neglected, may be solved by suc- 

 cessive approximation. 



As a first approximation the method of Section 

 6.5.2 is applied to determine points (h^jd) on the lobe. 

 The corresponding values of ii and v determine s 

 in Figure 19 or Figure 20, in Chapter 5, and tan^ 

 may be found from Figure 24 in Chapter 5 for the 

 given transmitter height hi. An alternate method 

 is to calculate tan;/- from equations (73), (58), and 

 (60) in Chapter 5, which are 



di = sd, 



d^ 

 2ka' 



h' = h 



tan\t = ^ii^. 

 rfi 



The angles i/'^ and v required to calculate the antenna 



pattern factors Fi and F^ are found from, equations 



(62) and (63) in Chapter 5, 



hi — hi d 



d 2ka 



tam^-d = 



5=0-1- 2«i {(j) = 180°, (j)' = 0), 



(23) 



v^4' + 



ha 



